<p>Numbers from 1 to 1000 are divisible by 60 but not by 24 is ___.</p>
Step-by-Step Solution
Key Concept: Find numbers divisible by 60, then subtract those divisible by both 60 and 24 (i.e., divisible by lcm(60,24) = 120). The key is recognizing that 'divisible by 60 but not by 24' means divisible by 60 AND not divisible by 24.
<p><strong>Step 1:</strong> Find count of numbers from 1 to 1000 divisible by 60.</p><p>⌊1000/60⌋ = 16 numbers</p><p><strong>Step 2:</strong> Find the condition 'divisible by 60 but not by 24'. A number divisible by both 60 and 24 must be divisible by lcm(60,24).</p><p>60 = 2² × 3 × 5</p><p>24 = 2³ × 3</p><p>lcm(60,24) = 2³ × 3 × 5 = 120</p><p><strong>Step 3:</strong> Find numbers divisible by both 60 and 24 (i.e., divisible by 120).</p><p>⌊1000/120⌋ = 8 numbers</p><p><strong>Step 4:</strong> Apply the exclusion principle.</p><p>Numbers divisible by 60 but NOT by 24 = 16 - 8 = 8</p><p>∴ Answer: <strong>8</strong></p>
Correct Answer: 8